# Finite square well bound states

• I
• andrewtz98
In summary, the finite potential well with regions of infinite and finite potential can be solved using the time-independent Schrodinger equation and the continuity conditions of Psi and its derivative. The relation between V0 and the number of bound states can be calculated graphically or numerically, with a possible approximate solution being V0 = (N-1)^2 * (h^2 / 32ma^2).

#### andrewtz98

Let's suppose I have a finite potential well: $$V(x)= \begin{cases} \infty,\quad x<0\\ 0,\quad 0<x<a\\ V_o,\quad x>a. \end{cases}$$

I solved the time-independent Schrodinger equation for each region and after applying the continuity conditions of ##\Psi## and its derivative I ended up with:

$$\tan(k_1a)=-\frac{k_1}{k_2},$$ where ##k_1=\sqrt{\frac{2mE}{\hbar^2}}## and ##k_2=\sqrt{\frac{2m(V_o-E)}{\hbar^2}}##.

I'm aware of the fact that solutions can only be calculated graphically, but what's the relation between the value of ##V_o## and the bound states? What if I want to find the acceptable values of ##V_o## for the bound states to be ##1,2,3,\dots## or none?

andrewtz98 said:
What if I want to find...
andrewtz98 said:
I'm aware of the fact that solutions can only be calculated graphically

$$V_0 \approx (N-1)^2 \frac{h^2}{32ma^2}$$